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Generalized propagator

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I don't understand how to calculate this generalized two-point function or propagator, used in some advanced topics in quantum field theory, a normal ordered product (denoted between ::) is subtracted from the usual time ordered product (denoted T):

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which is proportional to the identity operator ${\bf 1}$ by assumption. Now sandwich eq. (4) between the bra $\langle 0 |$ and the ket $|0\rangle $. Since the rhs. is proportional to the identity operator ${\bf 1}$, the unsandwiched rhs. must be equal to the sandwiched rhs. times the identity operator ${\bf 1}$. Hence also the unsandwiched lhs. must be equal to the sandwiched lhs. times the identity operator ${\bf 1}$. This yields eq. (1).

How do you get away without sandwiching on the left side of (4)? I would expect $\langle 0|T(X_i(t)X_j(t^\prime))|0\rangle ~-~\langle 0|:X_i(t)X_j(t^\prime):|0\rangle~=~\langle 0 | T(X_i(t)X_j(t^\prime))|0\rangle$ (which of course is a consequence of $\langle 0|:XX:|0\rangle = 0$).

This post imported from StackExchange Physics at 2014-03-17 03:36 (UCT), posted by SE-user David Z

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